We present a new framework to study the stability of semantic projections based on word embeddings. Roughly speaking, semantic projections are indices taking values in the interval $[0,1]$ that measure how terms share contextual meaning with the words of a given universe. Since there are many ways to define such projections, it is important to establish a procedure for verifying whether a group of them behaves similarly. Moreover, when fixing one particular projection, it is important to assess whether the average projections remain consistent when replacing the original universe with a similar one describing the same semantic environment. The aim of this paper is to address the lack of formal tools for assessing the stability of semantic projections (that is, their invariance under formal changes which preserve the underlying semantic context) across alternative but semantically related universes in word embedding models. To address these problems, we employ a combination of statistical and AI methods, including correlation analysis, clustering, chi-squared distance measures, weighted approximations, and Lipschitz-based estimators. The methodology provides theoretical guarantees under mild mathematical assumptions, ensuring bounded errors in projection estimations based on the assumption of Lipschitz continuity. We demonstrate the practical applicability of our approach through two case studies involving agricultural terminology across multiple data sources (DOAJ, Scholar, Google, and Arxiv). Our results show that semantic stability can be quantitatively evaluated and that the careful modeling of projection functions and universes is crucial for robust semantic analysis in NLP.

This article aims to expand the practical potential of the Chi and Chi-square distributions by extending their applicability to non-integer degrees of freedom, $ k $. The study has two main objectives: first, to explore and leverage the relationship between the Chi, Chi-square, and Gamma distributions to interpret non-integer degrees of freedom; and second, to apply this framework to real survival-time data in the context of biological and health sciences. Three specific cases are analyzed: the germination times of Pinus tropicalis Morelet seeds cultivated in western Cuba; the interval between infection by virulent tuberculous bacilli and death from tuberculosis in guinea pigs; and the period from the onset of symptoms to death due to COVID-19 in patients in Mexico during 2020. A parameter estimation was performed by maximum likelihood (MLE), and the optimal value of $ k $ was found to be non-integer in all cases.

Index spaces serve as valuable metric models for studying properties relevant to various applications, such as social science or economics. These properties are represented by real Lipschitz functions that describe the degree of association with each element within the underlying metric space. After determining the index value within a given sample subset, the classic McShane and Whitney formulas allow a Lipschitz regression procedure to be performed to extend the index values over the entire metric space. To improve the adaptability of the metric model to specific scenarios, this paper introduces the concept of a composition metric, which involves composing a metric with an increasing, positive and subadditive function $\varphi$. The results presented here extend well-established results for Lipschitz indices on metric spaces to composition metrics. In addition, we establish the corresponding approximation properties that facilitate the use of this functional structure. To illustrate the power and simplicity of this mathematical framework, we provide a concrete application involving the modeling of livability indices in North American cities.

Index spaces serve as valuable metric models for studying properties relevant to various applications, such as social science or economics. These properties are represented by real Lipschitz functions that describe the degree of association with each element within the underlying metric space. After determining the index value within a given sample subset, the classic McShane and Whitney formulas allow a Lipschitz regression procedure to be performed to extend the index values over the entire metric space. To improve the adaptability of the metric model to specific scenarios, this paper introduces the concept of a composition metric, which involves composing a metric with an increasing, positive and subadditive function $φ$. The results presented here extend well-established results for Lipschitz indices on metric spaces to composition metrics. In addition, we establish the corresponding approximation properties that facilitate the use of this functional structure. To illustrate the power and simplicity of this mathematical framework, we provide a concrete application involving the modelling of livability indices in North American cities.

Lipschitz-based classification provides a flexible framework for general metric spaces, naturally adapting to complex data structures without assuming linearity. However, direct applications of classical extensions often yield decision boundaries equivalent to the 1-Nearest Neighbour classifier, leading to overfitting and sensitivity to noise. Addressing this limitation, this paper introduces a novel binary classification algorithm that integrates probabilistic kernel smoothing with explicit Lipschitz extensions. We approximate the conditional probability of class membership by extending smoothed labels through a family of bounded Lipschitz functions. Theoretically, we prove that while direct extensions of binary labels collapse to nearest-neighbour rules, our probabilistic approach guarantees controlled complexity and stability. Experimentally, evaluations on synthetic and real-world datasets demonstrate that this methodology generates smooth, interpretable decision boundaries resilient to outliers. The results confirm that combining kernel smoothing with adaptive Lipschitz extensions yields performance competitive with state-of-the-art methods while offering superior geometric interpretability.